Finite difference approximations for a size-structured population model with distributed states in the recruitment

Citation

Ackleh AS, Farkas JZ, Li X & Ma B (2015) Finite difference approximations for a size-structured population model with distributed states in the recruitment. Journal of Biological Dynamics, 9 (Supplement 1), pp. 2-31. https://doi.org/10.1080/17513758.2014.923117

Abstract
We consider a size-structured population model where individuals may be recruited into the population at different sizes. First- and second-order finite difference schemes are developed to approximate the solution of the model. The convergence of the approximations to a unique weak solution is proved. We then show that as the distribution of the new recruits become concentrated at the smallest size, the weak solution of the distributed states-at-birth model converges to the weak solution of the classical Gurtin-McCamy-type size-structured model in the weak* topology. Numerical simulations are provided to demonstrate the achievement of the desired accuracy of the two methods for smooth solutions as well as the superior performance of the second-order method in resolving solution-discontinuities. Finally, we provide an example where supercritical Hopf-bifurcation occurs in the limiting single state-at-birth model and we apply the second-order numerical scheme to show that such bifurcation also occurs in the distributed model.

Keywords
continuous structured population models; distributed states-at-birth; finite difference approximations; convergence theory; existence and uniqueness of solutions

Journal
Journal of Biological Dynamics: Volume 9, Issue Supplement 1